Quadratic principal indecomposable modules and strongly real elements of finite Groups

TL;DR

This paper characterizes when principal indecomposable modules in characteristic 2 admit a non-degenerate quadratic form, linking this to the existence of certain involutions and strongly real elements in the group, and relates module counts to conjugacy class properties.

Contribution

It provides a criterion for quadratic forms on principal indecomposable modules in characteristic 2, connecting module theory with group involutions and conjugacy class analysis.

Findings

Characterization of quadratic principal indecomposable modules via involutions

Equivalence between module counts and strongly real conjugacy classes

Link between algebraic properties of Brauer characters and group elements

Abstract

Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic form if and only if there are involutions $s,t\in G$ such that $st$ has odd order and $\varphi(st)/2$ is not an algebraic integer. We then show that the number of isomorphism classes of quadratic principal indecomposable $G$-modules is equal to the number of strongly real conjugacy classes of odd order elements of $G$.